↑

Properties

Label 2-675-225.196-c1-0-24
Degree $2$
Conductor $675$
Sign $-0.968 + 0.247i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.956 − 1.06i)2-s + (−0.00470 + 0.0447i)4-s + (2.07 − 0.840i)5-s + (1.92 − 3.32i)7-s + (−2.26 + 1.64i)8-s + (−2.87 − 1.39i)10-s + (−3.49 − 3.88i)11-s + (−2.25 + 2.49i)13-s + (−5.37 + 1.14i)14-s + (3.99 + 0.849i)16-s + (3.90 − 2.84i)17-s + (−1.90 + 1.38i)19-s + (0.0278 + 0.0966i)20-s + (−0.781 + 7.43i)22-s + (−4.56 + 0.969i)23-s + ⋯
L(s)  = 1  + (−0.676 − 0.751i)2-s + (−0.00235 + 0.0223i)4-s + (0.926 − 0.375i)5-s + (0.725 − 1.25i)7-s + (−0.799 + 0.580i)8-s + (−0.909 − 0.442i)10-s + (−1.05 − 1.17i)11-s + (−0.624 + 0.693i)13-s + (−1.43 + 0.305i)14-s + (0.999 + 0.212i)16-s + (0.948 − 0.688i)17-s + (−0.437 + 0.318i)19-s + (0.00623 + 0.0216i)20-s + (−0.166 + 1.58i)22-s + (−0.951 + 0.202i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.968 + 0.247i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.968 + 0.247i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130445 - 1.03864i\)
\(L(\frac12)\) \(\approx\) \(0.130445 - 1.03864i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.07 + 0.840i)T \)
good2 \( 1 + (0.956 + 1.06i)T + (-0.209 + 1.98i)T^{2} \)
7 \( 1 + (-1.92 + 3.32i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.49 + 3.88i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (2.25 - 2.49i)T + (-1.35 - 12.9i)T^{2} \)
17 \( 1 + (-3.90 + 2.84i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.90 - 1.38i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.56 - 0.969i)T + (21.0 - 9.35i)T^{2} \)
29 \( 1 + (-6.71 + 2.98i)T + (19.4 - 21.5i)T^{2} \)
31 \( 1 + (-0.593 - 0.264i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (-0.315 - 0.970i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.50 + 2.78i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (2.34 - 4.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (10.9 - 4.85i)T + (31.4 - 34.9i)T^{2} \)
53 \( 1 + (2.92 + 2.12i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-5.20 + 5.78i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (4.40 + 4.89i)T + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (0.930 + 0.414i)T + (44.8 + 49.7i)T^{2} \)
71 \( 1 + (-2.82 - 2.04i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.38 - 4.25i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.49 + 0.667i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (-1.18 - 11.2i)T + (-81.1 + 17.2i)T^{2} \)
89 \( 1 + (-0.935 + 2.87i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-14.8 + 6.59i)T + (64.9 - 72.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.02779856484946444088943974933, −9.660247951257377901289805629413, −8.389303128399238451041926830240, −7.88395656654032690813616531392, −6.47447307635857388202222980649, −5.49609298427104944601802056168, −4.61846137910301522687280959977, −3.01900897674326560402950762727, −1.83295438419874796790262637377, −0.68223477327672396977711882983, 2.05154344631875443339309539354, 2.95529268995831868633393175696, 4.89524482073747412417255754026, 5.65879884512134221355238477709, 6.52285785915694557962144841219, 7.60829470861862295966185512728, 8.188170238571720700409974930662, 9.026955532936853350637989225806, 10.03437728000635568877958231457, 10.37994236609057976488599776809

Graph of the $Z$-function along the critical line